\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  -1.0  &   & +  6.00 x_{2} & +  5.00 x_{3} & +  5.00 x_{4} & +  5.00 x_{5} &   \\
 x_{8}   &  1.0 & -9.00 x_{1} & + 10.00 x_{2} & +  5.00 x_{3} & -6.00 x_{4} &   & -6.00 x_{6}\\
 x_{9}   &  3.0 & +  6.00 x_{1} & -6.00 x_{2} & +  3.00 x_{3} &   & -9.00 x_{5} & +  8.00 x_{6}\\
 x_{10}   &  0.0 & +  8.00 x_{1} & +  3.00 x_{2} & +  3.00 x_{3} & + 10.00 x_{4} & +  7.00 x_{5} &   \\
 x_{11}   &  -2.0 & -8.00 x_{1} & -9.00 x_{2} & +  8.00 x_{3} & -6.00 x_{4} & + 10.00 x_{5} & +  5.00 x_{6}\\
 x_{12}   &  3.0 & -5.00 x_{1} & +  7.00 x_{2} & -3.00 x_{3} & -5.00 x_{4} & +  1.00 x_{5} & +  3.00 x_{6}\\
 x_{13}   &  1.0 & -9.00 x_{1} & +  1.00 x_{2} & +  4.00 x_{3} & +  9.00 x_{4} & +  7.00 x_{5} & + 10.00 x_{6}\\
\hline
z    &  0.0 & -2.00 x_{1} & +  3.00 x_{2} & +  4.00 x_{3} & -3.00 x_{4} & -4.00 x_{5} & -1.00 x_{6}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{6}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,13$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0  &   & +  9.00 y_{8} & -6.00 y_{9} & -8.00 y_{10} & +  8.00 y_{11} & +  5.00 y_{12} & +  9.00 y_{13}\\
 y_{2}   &  1.0 & -6.00 y_{7} & -10.00 y_{8} & +  6.00 y_{9} & -3.00 y_{10} & +  9.00 y_{11} & -7.00 y_{12} & -1.00 y_{13}\\
 y_{3}   &  1.0 & -5.00 y_{7} & -5.00 y_{8} & -3.00 y_{9} & -3.00 y_{10} & -8.00 y_{11} & +  3.00 y_{12} & -4.00 y_{13}\\
 y_{4}   &  1.0 & -5.00 y_{7} & +  6.00 y_{8} &   & -10.00 y_{10} & +  6.00 y_{11} & +  5.00 y_{12} & -9.00 y_{13}\\
 y_{5}   &  1.0 & -5.00 y_{7} &   & +  9.00 y_{9} & -7.00 y_{10} & -10.00 y_{11} & -1.00 y_{12} & -7.00 y_{13}\\
 y_{6}   &  1.0  &   & +  6.00 y_{8} & -8.00 y_{9} &   & -5.00 y_{11} & -3.00 y_{12} & -10.00 y_{13}\\
\hline
z    &  -0 & +  1.00 y_{7} & -1.00 y_{8} & -3.00 y_{9} &   & +  2.00 y_{11} & -3.00 y_{12} & -1.00 y_{13}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 3 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.07619047619 & +  0.38 y_{2} & + 12.81 y_{8} & -4.17 y_{9} & -10.06 y_{10} & -0.46 y_{5} & +  7.21 y_{12} & +  6.18 y_{13}\\
 y_{7}   &  0.180952380952 & -0.10 y_{2} & -0.95 y_{8} & +  1.34 y_{9} & -0.89 y_{10} & -0.09 y_{5} & -0.75 y_{12} & -0.70 y_{13}\\
 y_{3}   &  0.0190476190476 & +  0.10 y_{2} & -4.05 y_{8} & -11.54 y_{9} & +  3.49 y_{10} & +  0.89 y_{5} & +  4.55 y_{12} & +  2.30 y_{13}\\
 y_{4}   &  0.152380952381 & +  0.76 y_{2} & + 13.62 y_{8} & -5.34 y_{9} & -7.11 y_{10} & +  0.09 y_{5} & + 10.42 y_{12} & -7.64 y_{13}\\
 y_{11}   &  0.00952380952381 & +  0.05 y_{2} & +  0.48 y_{8} & +  0.23 y_{9} & -0.26 y_{10} & -0.06 y_{5} & +  0.28 y_{12} & -0.35 y_{13}\\
 y_{6}   &  0.952380952381 & -0.24 y_{2} & +  3.62 y_{8} & -9.14 y_{9} & +  1.29 y_{10} & +  0.29 y_{5} & -4.38 y_{12} & -8.24 y_{13}\\
\hline
z    &  0.2  &   & -1.00 y_{8} & -1.20 y_{9} & -1.40 y_{10} & -0.20 y_{5} & -3.20 y_{12} & -2.40 y_{13}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{2}   &  -0 & -0.38 x_{1} & +  0.10 x_{7} & -0.10 x_{3} & -0.76 x_{4} & -0.05 x_{11} & +  0.24 x_{6}\\
 x_{8}   &  1.0 & -12.81 x_{1} & +  0.95 x_{7} & +  4.05 x_{3} & -13.62 x_{4} & -0.48 x_{11} & -3.62 x_{6}\\
 x_{9}   &  1.2 & +  4.17 x_{1} & -1.34 x_{7} & + 11.54 x_{3} & +  5.34 x_{4} & -0.23 x_{11} & +  9.14 x_{6}\\
 x_{10}   &  1.4 & + 10.06 x_{1} & +  0.89 x_{7} & -3.49 x_{3} & +  7.11 x_{4} & +  0.26 x_{11} & -1.29 x_{6}\\
 x_{5}   &  0.2 & +  0.46 x_{1} & +  0.09 x_{7} & -0.89 x_{3} & -0.09 x_{4} & +  0.06 x_{11} & -0.29 x_{6}\\
 x_{12}   &  3.2 & -7.21 x_{1} & +  0.75 x_{7} & -4.55 x_{3} & -10.42 x_{4} & -0.28 x_{11} & +  4.38 x_{6}\\
 x_{13}   &  2.4 & -6.18 x_{1} & +  0.70 x_{7} & -2.30 x_{3} & +  7.64 x_{4} & +  0.35 x_{11} & +  8.24 x_{6}\\
\hline
z    &  -0.2 & -1.08 x_{1} & -0.18 x_{7} & -0.02 x_{3} & -0.15 x_{4} & -0.01 x_{11} & -0.95 x_{6}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{2}   &  -0 & -0.38 x_{1} & +  0.10 x_{7} & -0.10 x_{3} & -0.76 x_{4} & -0.05 x_{11} & +  0.24 x_{6}\\
 x_{8}   &  1.0 & -12.81 x_{1} & +  0.95 x_{7} & +  4.05 x_{3} & -13.62 x_{4} & -0.48 x_{11} & -3.62 x_{6}\\
 x_{9}   &  1.2 & +  4.17 x_{1} & -1.34 x_{7} & + 11.54 x_{3} & +  5.34 x_{4} & -0.23 x_{11} & +  9.14 x_{6}\\
 x_{10}   &  1.4 & + 10.06 x_{1} & +  0.89 x_{7} & -3.49 x_{3} & +  7.11 x_{4} & +  0.26 x_{11} & -1.29 x_{6}\\
 x_{5}   &  0.2 & +  0.46 x_{1} & +  0.09 x_{7} & -0.89 x_{3} & -0.09 x_{4} & +  0.06 x_{11} & -0.29 x_{6}\\
 x_{12}   &  3.2 & -7.21 x_{1} & +  0.75 x_{7} & -4.55 x_{3} & -10.42 x_{4} & -0.28 x_{11} & +  4.38 x_{6}\\
 x_{13}   &  2.4 & -6.18 x_{1} & +  0.70 x_{7} & -2.30 x_{3} & +  7.64 x_{4} & +  0.35 x_{11} & +  8.24 x_{6}\\
\hline
z    &  -0.8 & -4.97 x_{1} & -0.06 x_{7} & +  7.26 x_{3} & -4.94 x_{4} & -0.37 x_{11} & +  0.86 x_{6}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{2}   &  -0 & -0.38 x_{1} & +  0.10 x_{7} & -0.10 x_{3} & -0.76 x_{4} & -0.05 x_{11} & +  0.24 x_{6}\\
 x_{8}   &  1.0 & -12.81 x_{1} & +  0.95 x_{7} & +  4.05 x_{3} & -13.62 x_{4} & -0.48 x_{11} & -3.62 x_{6}\\
 x_{9}   &  1.2 & +  4.17 x_{1} & -1.34 x_{7} & + 11.54 x_{3} & +  5.34 x_{4} & -0.23 x_{11} & +  9.14 x_{6}\\
 x_{10}   &  1.4 & + 10.06 x_{1} & +  0.89 x_{7} & -3.49 x_{3} & +  7.11 x_{4} & +  0.26 x_{11} & -1.29 x_{6}\\
 x_{5}   &  0.2 & +  0.46 x_{1} & +  0.09 x_{7} & -0.89 x_{3} & -0.09 x_{4} & +  0.06 x_{11} & -0.29 x_{6}\\
 x_{12}   &  3.2 & -7.21 x_{1} & +  0.75 x_{7} & -4.55 x_{3} & -10.42 x_{4} & -0.28 x_{11} & +  4.38 x_{6}\\
 x_{13}   &  2.4 & -6.18 x_{1} & +  0.70 x_{7} & -2.30 x_{3} & +  7.64 x_{4} & +  0.35 x_{11} & +  8.24 x_{6}\\
\hline
z    &  -0.8 & -4.97 x_{1} & -0.06 x_{7} & +  7.26 x_{3} & -4.94 x_{4} & -0.37 x_{11} & +  0.86 x_{6}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{2} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  -0 & -4.00 x_{1} & +  1.00 x_{7} & -10.50 x_{2} & -8.00 x_{4} & -0.50 x_{11} & +  2.50 x_{6}\\
 x_{8}   &  1.0 & -29.00 x_{1} & +  5.00 x_{7} & -42.50 x_{2} & -46.00 x_{4} & -2.50 x_{11} & +  6.50 x_{6}\\
 x_{9}   &  1.2 & -42.00 x_{1} & + 10.20 x_{7} & -121.20 x_{2} & -87.00 x_{4} & -6.00 x_{11} & + 38.00 x_{6}\\
 x_{10}   &  1.4 & + 24.00 x_{1} & -2.60 x_{7} & + 36.60 x_{2} & + 35.00 x_{4} & +  2.00 x_{11} & -10.00 x_{6}\\
 x_{5}   &  0.2 & +  4.00 x_{1} & -0.80 x_{7} & +  9.30 x_{2} & +  7.00 x_{4} & +  0.50 x_{11} & -2.50 x_{6}\\
 x_{12}   &  3.2 & + 11.00 x_{1} & -3.80 x_{7} & + 47.80 x_{2} & + 26.00 x_{4} & +  2.00 x_{11} & -7.00 x_{6}\\
 x_{13}   &  2.4 & +  3.00 x_{1} & -1.60 x_{7} & + 24.10 x_{2} & + 26.00 x_{4} & +  1.50 x_{11} & +  2.50 x_{6}\\
\hline
z    &  -0.8 & -34.00 x_{1} & +  7.20 x_{7} & -76.20 x_{2} & -63.00 x_{4} & -4.00 x_{11} & + 19.00 x_{6}\\
\end{array}\]


 $ x_{6} $ enters and $ x_{5} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  0.2  &   & +  0.20 x_{7} & -1.20 x_{2} & -1.00 x_{4} &   & -1.00 x_{5}\\
 x_{8}   &  1.52 & -18.60 x_{1} & +  2.92 x_{7} & -18.32 x_{2} & -27.80 x_{4} & -1.20 x_{11} & -2.60 x_{5}\\
 x_{9}   &  4.24 & + 18.80 x_{1} & -1.96 x_{7} & + 20.16 x_{2} & + 19.40 x_{4} & +  1.60 x_{11} & -15.20 x_{5}\\
 x_{10}   &  0.6 & +  8.00 x_{1} & +  0.60 x_{7} & -0.60 x_{2} & +  7.00 x_{4} & -0.00 x_{11} & +  4.00 x_{5}\\
 x_{6}   &  0.08 & +  1.60 x_{1} & -0.32 x_{7} & +  3.72 x_{2} & +  2.80 x_{4} & +  0.20 x_{11} & -0.40 x_{5}\\
 x_{12}   &  2.64 & -0.20 x_{1} & -1.56 x_{7} & + 21.76 x_{2} & +  6.40 x_{4} & +  0.60 x_{11} & +  2.80 x_{5}\\
 x_{13}   &  2.6 & +  7.00 x_{1} & -2.40 x_{7} & + 33.40 x_{2} & + 33.00 x_{4} & +  2.00 x_{11} & -1.00 x_{5}\\
\hline
z    &  0.72 & -3.60 x_{1} & +  1.12 x_{7} & -5.52 x_{2} & -9.80 x_{4} & -0.20 x_{11} & -7.60 x_{5}\\
\end{array}\]


 $ x_{7} $ enters and $ x_{6} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  0.25 & +  1.00 x_{1} & -0.63 x_{6} & +  1.13 x_{2} & +  0.75 x_{4} & +  0.13 x_{11} & -1.25 x_{5}\\
 x_{8}   &  2.25 & -4.00 x_{1} & -9.13 x_{6} & + 15.63 x_{2} & -2.25 x_{4} & +  0.63 x_{11} & -6.25 x_{5}\\
 x_{9}   &  3.75 & +  9.00 x_{1} & +  6.12 x_{6} & -2.63 x_{2} & +  2.25 x_{4} & +  0.38 x_{11} & -12.75 x_{5}\\
 x_{10}   &  0.75 & + 11.00 x_{1} & -1.87 x_{6} & +  6.37 x_{2} & + 12.25 x_{4} & +  0.37 x_{11} & +  3.25 x_{5}\\
 x_{7}   &  0.25 & +  5.00 x_{1} & -3.13 x_{6} & + 11.63 x_{2} & +  8.75 x_{4} & +  0.63 x_{11} & -1.25 x_{5}\\
 x_{12}   &  2.25 & -8.00 x_{1} & +  4.88 x_{6} & +  3.62 x_{2} & -7.25 x_{4} & -0.38 x_{11} & +  4.75 x_{5}\\
 x_{13}   &  2.0 & -5.00 x_{1} & +  7.50 x_{6} & +  5.50 x_{2} & + 12.00 x_{4} & +  0.50 x_{11} & +  2.00 x_{5}\\
\hline
z    &  1.0 & +  2.00 x_{1} & -3.50 x_{6} & +  7.50 x_{2} & +  0.00 x_{4} & +  0.50 x_{11} & -9.00 x_{5}\\
\end{array}\]


 $ x_{1} $ enters and $ x_{12} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  0.53125 & -0.13 x_{12} & -0.02 x_{6} & +  1.58 x_{2} & -0.16 x_{4} & +  0.08 x_{11} & -0.66 x_{5}\\
 x_{8}   &  1.125 & +  0.50 x_{12} & -11.56 x_{6} & + 13.81 x_{2} & +  1.37 x_{4} & +  0.81 x_{11} & -8.62 x_{5}\\
 x_{9}   &  6.28125 & -1.13 x_{12} & + 11.61 x_{6} & +  1.45 x_{2} & -5.91 x_{4} & -0.05 x_{11} & -7.41 x_{5}\\
 x_{10}   &  3.84375 & -1.37 x_{12} & +  4.83 x_{6} & + 11.36 x_{2} & +  2.28 x_{4} & -0.14 x_{11} & +  9.78 x_{5}\\
 x_{7}   &  1.65625 & -0.62 x_{12} & -0.08 x_{6} & + 13.89 x_{2} & +  4.22 x_{4} & +  0.39 x_{11} & +  1.72 x_{5}\\
 x_{1}   &  0.28125 & -0.12 x_{12} & +  0.61 x_{6} & +  0.45 x_{2} & -0.91 x_{4} & -0.05 x_{11} & +  0.59 x_{5}\\
 x_{13}   &  0.59375 & +  0.62 x_{12} & +  4.45 x_{6} & +  3.23 x_{2} & + 16.53 x_{4} & +  0.73 x_{11} & -0.97 x_{5}\\
\hline
z    &  1.5625 & -0.25 x_{12} & -2.28 x_{6} & +  8.41 x_{2} & -1.81 x_{4} & +  0.41 x_{11} & -7.81 x_{5}\\
\end{array}\]


 $ x_{2} $ enters and Unbounded Dictionary!


 $ x_{2} $ enters and Unbounded Dictionary!
\end{document}
